Using a linear rule of the form y = ax + b on whole-number inputs, 7 is…
2026
Using a linear rule of the form y = ax + b on whole-number inputs, 7 is related to 31. Following the same rule, 9 is related to 43. To which of the following is 13 related? (Note: Perform operations on each whole number without splitting it into constituent digits. For example, you may add, subtract, or multiply 13 as a whole, but you may not separate it into 1 and 3.)
Answer: A. 67 — ConceptIn a number analogy, one consistent transformation must map every input to its paired output. For a linear rule y = ax + b, two given pairs determine a…
- A.
67
- B.
63
- C.
55
- D.
68
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Correct answer: A
Concept
In a number analogy, one consistent transformation must map every input to its paired output.
For a linear rule y = ax + b, two given pairs determine a and b by comparing their equations.
Application
Let the rule be y = ax + b. The pairs (7, 31) and (9, 43) give 31 = 7a + b and 43 = 9a + b.
Subtracting the first equation from the second gives 12 = 2a, so a = 6.
Substituting a = 6 into 31 = 7a + b gives b = 31 − 42 = −11.
Therefore, for x = 13, y = 6 × 13 − 11 = 78 − 11 = 67.
Cross-check
The same rule reproduces both examples: 6 × 7 − 11 = 31 and 6 × 9 − 11 = 43.
Thus, 13 is related to 67.